WEB/ONLINE AT&T MESSAGE APP ONLY SHOWS MY TEXT MESSAGES, NOT THE ENTIRECONVERSATION 😡

My husband and I switched from Verizon to AT&T less than two weeks ago and when texting someone from our cell phones, only his/my messages show up on messages.att.net (AT&T messages). So when we move from cell phone to PC/tablet/iPad we still have to look back at the cell phone to view all what has been texted. As big as AT&T is and for as long as they have been in business, WHY this issue? I see that this issue has been posted about since 2013. Am I overlooking any solutions to this? We never had any problems when we used the Verizon app on his tablet and my iPad or the PC. We aren't going to take our phones back to AT&T for a partial refund after the restocking fee and return to Verizon as it really is too much of a hazzle but after our cell phones are paid off in two years, if the problem still remains we will be returning to Verizon. I do regret that I didn't picked phones that were unlocked so we could have taken them to Verizon.